Let \(\Phi := \lbrace w.\alpha _i|(w,i)\in W^{v}\times I\rbrace \) (resp. \(\Phi ^\vee =\lbrace w.\alpha _i^\vee |(w,i)\in W^{v}\times I\rbrace \) ) be the set of real roots (resp. real coroots): then \(\Phi \)  (resp. \(\Phi ^\vee \) ) is a subset of the root lattice \(Q_\mathbb {Z}:= \displaystyle \bigoplus _{i\in I}\mathbb {Z}\alpha _i\)  (resp. coroot lattice \(Q^\vee _\mathbb {Z}=\bigoplus _{i\in I}\mathbb {Z}\alpha _i^\vee \) ). By [1]}, we have \(\mathbb {R}\alpha ^\vee \cap \Phi ^\vee =\lbrace \pm \alpha ^\vee \rbrace \)  and \(\mathbb {R}\alpha \cap \Phi =\lbrace \pm \alpha \rbrace \)  for all \(\alpha ^\vee \in \Phi ^\vee \)  and \(\alpha \in \Phi \) .
